I 895-] BAKER DIRECTED MAGNITUDES. I 63 



If we take the second manner of difference as the process which 

 is to be repeated, we call it division, or the doing- to the operand 

 whatever was done to the operator to produce unity. 



I I. 



Hitherto we have considered the various positions composing a 

 point as merely bunched together, but if we consider these positions as 

 arranged in a certain order or sense, then a point has in addition to 

 its weight the characteristic of normalcy or opposition of the weight. 

 The normal scmse or direction of arrangement we designate by +, the 

 reverse sense or direction by — . 



In addition to the normal and reversed senses of the weight, we 

 have also the mean reversed sense indicated by ]/ (see Algebraic 

 Symbols, Am. Jour. Math., xviii., 62). 



The weights, having merely magnitude (with sense) are scalars, 

 that is, magnitudes whose properties can be scaled off on a scale. 



Out of the weights of the points, then, we get six fundamental 

 operations : addition, subtraction, multiplication, division, reversion 

 and mean reversion — and no others. 



III. 



Applying these six operations to scalars, the only magnitude so 

 far, we get the following results : 



The sum of two scalars is a scalar ; 



The difference of two scalars is a scalar ; 



The product of two scalars is a scalar ; 



The quotient of two scalars is a scalar ; 



The reversion of a scalar is a scalar ; 



The mean reversion of a scalar is a vector (loc. cit. ), a quantity 

 having magnitude and direction. 



Applying the six algebraic operations to this new quantity, the 

 vector,' we get : 



The sum of two vectors is a vector ; 



The difference of two vectors is a vector ; 



The product of two vectors is a quater?iio7i (a scalar plus a vector, 

 a scalar, or a vector, according to the position of the mean reversing 

 operator or symbol) (loc. cit.); 



The quotient of two vectors is a quaternion ; 



The reversion of a vector is a vector ; 



The mean reversion of a vector is a quaternion. 



